Molecular electron density contours

Fig. 1.32. (a) Molecular graphs and electron density contours for pentane and hexane. Dots on bond paths represent critical points, (b) Comparison of molecular graphs for bicycloalkanes and corresponding propellanes. (Reproduced from Chem. Rev. 91 893 (1991) with permission of the American Chemical Society.)... 

Molecular orbitals and electron density contours formed from the combination... 

We have shown in earlier work that it is possible to quantitatively relate a variety of liquid, solid and solution phase properties to the electrostatic potential patterns on the surfaces of the individual molecules [64-66]. Among these properties are pKa, boiling points and critical constants, enthalpies of fusion, vaporization and sublimation, solubilities, partition coefficients, diffusion constants and viscosities. For these purposes, we take the molecular surface to be the 0.001 au contour of the molecular electronic density p(r), following the suggestion of Bader et al [67],... 

The molecular volume descriptor, V, can be recognized as an important descriptor once one realizes that the free energy of solution is related in part to the size of the cavity that must be carved out of the solvent bath by the solute molecule during the solvation process. The surface area, A, of a molecule or a fragment of a molecule may be construed as a measure of the region available for interaction with another molecule. For computing V and A, one could use a particular electron density contour or a non-QM-derived measure of atomic size such as the van der Waals radii available from standard tables in physical chemistry textbooks. 

Shape complementarity of molecular electron densities represented by MIDCO s involves complementary curvatures, as well as complementary values of the charge density contour parameters a. In general, a locally convex domain relative to a reference curvature b shows shape complementarity with a locally concave domain relative to a reference curvature -b. Furthermore, shape complementarity between the lower electron density contours of one molecule and the higher electron density contours of the other molecule is of importance. 

Such an expression has previously been used for comparative purposes, for the study of interaction between two molecular species, by computing the electrostatic potential of the first partner and by assuming some point charge model as representative of the charge distribution of the second partner. We also plan to extend this concept in a more subtle way by using an electron density contour map to describe the charge distribution of the second partner as a function of the space surrounding this second partner. 

However, our preference is to follow the suggestion of Bader et al. [86] and take the surface to be an outer contour of the molecular electronic density. This has the important advantage that it reflects features specific to the particular molecule, such as lone pairs or strained bonds. We normally choose p(r) = 0.001 electrons/bohr3, but we have confirmed that other low values of p(r), e.g., 0.002 electrons/bohr3, would serve equally well [87]. 

While this result confirmed the feasibility of the general approach, it did not precipitate wider exploration of dielectric medium effects. Recently, however, Wiberg et al. have incorporated the Onsager self-consistent reaction-field model into ab initio MO theory in an implementation which provides analytical gradients and second derivatives. The model considers just the dipole of the solute molecules and a spherical cavity whose radius is chosen for a given solute molecule from the molecular volume estimated at the 0.001 eB electron-density contour (B is the Bohr radius), plus an empirical constant 0.5 A to account for the nearest approach of solvent molecules [164]. Cieplak and Wiberg have used this model to probe solvent effects on the transition states for nucleophilic additions to substituted acetaldehydes [165]. For each... 

The critical point is the point at which the gradient vector field for the charge density is zero, that is, either a maximum or minimum along N. The condition Vp(r) N(r) = 0 applied to other paths between two atoms defines a unique surface that can represent the boundary of the atoms within the molecule. The electron density within these boundaries then gives the atomic charge. The combination of electron density contours, bond paths, and critical points defines the molecular graph. This analysis can be applied to electron density calculated by either MO or DFT methods. For a very simple molecule such as Hj, the bond path is a straight line between the nuclei. The... 

The molecular volume can in turn be obtained by dividing the molecular weight by the density or from refractivity measurements is Avogadro s number. The cavity radius can also be estimated from the largest interatomic distance within the molecule. A third approach is to calculate the volume of the molecule from a suitable electron density contour. The radii obtained by these procedures are often adjusted by adding an empirical constant to give the true cavity radius. This extra value accounts for the fact that solvent... 

Admitting the existence of a confined molecular space, one can in principle approximate it by contour surfaces of the molecular electron density function p(r). Values of the isodensity contour levels that yield volumes most consistent with experiments are given in the literature. 3 Moreover, from... 

FIGURE 13.20 Hartree-Fock MO electron-density contours for the ground electronic state of Li2 as calculated by Wahl. [A. CWahl,.Sctcwcc, 151,961 (l%6y, Scientific American, April l9J0,p.54 Atomic and Molecular Structure-. 4 Wall Charts, McGraw-Hill, 1970.]... 

Figure Z5 Electron density contour diagrams for HjO (a) in the molecular plane and (b) in the perpendicular plane from the ah initio SCF calculation given in [14]. (Redrawn from [14] with permission from the American Chemical Society.)...

Electron density contour surfaces have the mathematical property of compactness, a generalization of the properties of closed and bounded . The Hausdorff distance h(A, B) between two (compact) subsets A and B of X is defined as the lowest upper bound h(A, B) = sup g gg(rf(a, B), d b, A) of distances between points a of A and the set B and distances between points b of B and the set A. In particular, the Hausdorff distance between two superimposed molecular contour surfaces (which are closed sets) is the minimum r value such that any point on either contour surface has at least one point of the other contour surface within a distance r. 

IlyperCl hem can display molecular orbitals and the electron density ol each molecular orbital as contour plots, showing the nodal structure and electron distribution in the molecular orbitals. 

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